Yes to all. And maybe.
Up here in New England in the dead of winter the ground is pretty much rock solid and a tank would sit on that as it would on a paved road. Lakes freeze solidly enough that ice fisherman will drive their SUV's out to their fishing shacks and I can recall in the late 60's having seen cars racing on a pond. The ground will be just as frozen.
Snow is like rain, only colder and more annoying! Just as you in Texas can have a light rain shower which only moistens the soil but does not create mud, snow can cover the ground and the ground would not be muddy.
Also, consider that typically, an inch of snow equals about 1/10th inch of rain, not much. Snow will accumulate on both frozen and non frozen soil, depending on how cold it has been before the snow fell. Just as you wouldn't expect to see a vehicle covered in mud after a passing rain shower, a lilght covering of snow will not necessarily result in a muddy field.
After a very cold and dry spell. you could very easily have an accumulation of snow in which a tank could leave tracks yet show no mud. On the other hand, you could have a cold snap after a wet warm period where the snow would accumulate, and the ground would be wet and moist and your hypothetical tank would leave tracks and be muddied.